When free-surface vortexing is relevant in stirred bioreactors, which dimensionless number is most useful to characterize the balance of inertial forces to gravity?

Introduction / Context:
Free-surface behavior in stirred tanks affects gas entrainment and mixing. A central measure for vortex formation is the Froude number, which compares inertial forces generated by rotation to gravitational forces that flatten the surface.

Given Data / Assumptions:

• Unbaffled or partially baffled tank with a free liquid surface.
• Impeller of diameter Di rotating at speed N.
• Newtonian liquid under normal gravity g.

Concept / Approach:
The impeller Froude number is Fr = N^2 * Di / g (or sometimes defined with tip speed squared over g * Di). Higher Fr implies stronger swirling relative to gravity and deeper vortices. Other dimensionless numbers (Stanton, Rayleigh, Bond) are used for different phenomena: Stanton for convective heat/mass transfer coefficients, Rayleigh for buoyancy-driven convection, and Bond for gravity vs surface tension for droplets/bubbles—none directly quantify vortexing due to mechanical agitation.

Step-by-Step Solution:
Identify phenomenon: vortex depth and stability at the free surface.Select ratio: inertial force (∝ N^2 * Di) to gravitational force (∝ g).Compute Fr: Fr = N^2 * Di / g.Use Fr to compare conditions and predict onset/intensity of vortexing.

Verification / Alternative check:
Empirical correlations show vortex depth normalized by liquid height increases with Fr, and baffling effectively reduces Fr-relevant swirling, diminishing vortex formation.

Why Other Options Are Wrong:
Stanton: relates h/(ρ * U * c_p) or mass-transfer analogs; not a vortexing metric.Rayleigh: buoyancy-driven convection in non-agitated systems.Bond: gravity vs surface tension in droplets/bubbles, not tank vortexing.

Common Pitfalls:
Using Reynolds number to predict vortexing (Re gauges turbulence, not free-surface deformation) or ignoring the role of liquid height and baffling.

Final Answer:
Froude number